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The Pope has no special way of talking to God. In Mat 6:9-13 Jesus taught his disciples how to talk to God, one of whom was Peter, the founder of the Papacy. However, everyone has direct access to God and able to talk to Him at anytime, anywhere.
As for the visionaries, you can be certain to discount every one of them. The Bible defines the standard of visions, which is concrete and specific. Vague claims that lead to nothing but belief in the apparition itself is not from God. Most visitations result in extreme fear, pralysis, and near death (Eze 1:28-2:3; Dan 10:7-11; Rev 1:17). Others appear normal but carry a specific purpose (Gen 18:9-15). If neither are present you should be very skeptical.
Upvote:1
But does the Pope directly talk with God or the Virgin Mary? Or, does he do it in any other, not-direct, way?
First, notice that God is Trinitarian, so anyone "speaking" to God must speak to one of the three persons, either God the Father, Jesus or the Holy Spirit.
Now, regarding visions or other forms of visitations, if we are to believe the apocryphal Acts of Peter, then Peter himself (the First pope) was the only Pope in history of which we have recollection to have directly talked to God (as Jesus). There seems to be no other account of such encounter.
Regarding talking to God in "other, not-direct, way", of course! That's what prayer is all about. Jesus taught us and encouraged us to pray to God the Father. He Himself gave us the example of such prayer. Thus, this is not limited to popes but open to every Christian. We can also direct our prayers/conversation to Jesus and the Holy Spirit.
Now, it is only us (so far as I know) who speak in terms of words to God. God might respond in multiple, sometimes unexpected ways, to our prayers and questions. Many times we are so busy and anxious with the daily tolls of life that there is no enough silence in our hearts/souls in order to listen to God.